Add all the ingredients to a saucepan; bring to a simmer over medium-high heat, stirring just until the sugar dissolves.
Decrease heat and simmer until the berries have softened, about 10 minutes.
Cover and cool off the heat for 30 minutes.
Pour the contents of the pan into a large blender; puree until smooth.
Strain through a fine-mesh sieve into a 13x9 inch baking pan, pushing the solids against the mesh with the back of a wooden spoon.
Freeze the mixture in the pan on the floor of the freezer for about 3 hours, stirring about every 20 minutes with a fork to break up the ice crystals and distribute them evenly throughout the pan.
After the mixture has frozen but is not yet fully firm, cover it tightly with plastic wrap; the granita will keep this way for up to 3 weeks.
Scrape the surface of the granita with a fork to produces lots of shaved ice crystals, scoop these into individual cups or glasses for serving.
*Spiced Cranberry Granita: add 1 teaspoon ground cinnamon, ¼ t. ground mace, and ¼ t. grated nutmeg to the pan before you set the mixture aside to cool for 20 minutes; proceed as directed.